Reflexive sheaves and resolutions on smooth projective surfaces

Question

Let $E$ be a coherent torsion free sheaf on a smooth projective surface $S$ over $\mathbf{C}$. Then I was wondering about the conditions for $E$ to be reflexive, i.e. the natural $E \rightarrow E^{**}$ is an iso.
I read in Hartshornes paper "stable reflexive sheaves" on p.125 that E torsion free is reflexive iff for all $x \in S$ with $\dim(\mathcal{O}_x)\geq 2$ we have $\text{depth}(E_x)\geq 2$.
Now by Auslaender-Buchsbaum, $\text{dh}(E_x)+\text{depth}(E_x)=\dim(\mathcal{O}_x)$ ,where $\text{dh}(E_x)$ is the homological dimension of $E_x$, i.e. the length of a projective resolution $E_x^\bullet \rightarrow E_x$. This resolution can chosen to be consisting of locally frees as $S$ is regular (standard fact).
Now combining these two theorems, $\text{dh}(E_x) = 0$ for all closed points in $S$, so $E$ itself must be locally free? Here I'm not 100% sure, where is my mistake?
Any help appreciated!!